SOLUTION: . In a group of 150 students, more students are taking a math class than are taking a science class this semester. If 80 are taking a science class and 25 are not taking either ma

There are four categories of students here:

1. Those taking math but not science.
2. Those taking science but not math.
3. Those taking both math and science.
4. Those taking neither math nor science.

1. Let x = the number of students taking math but not science.
2. Let y = the number of students taking science but not math.
3. Let z = the number of students taking both math and science.
4. Let w = the number of students taking neither math nor science.

We are asked:

>>.....what is the minimum number of students who could be 
taking both math and science this semester?.....<<

So we want to know the smallest possible value of z.

>>......In a group of 150 students,......<<

Therefore, x + y + z + w = 150 

>>......more students are taking a math class than are taking 
a science class this semester......<<

Therefore x + z > y + z.  Subtracting z from both sides,
              x > y

 
>>......If 80 are taking a science class......<<

Therefore y + z = 80

>>......and 25 are not taking either math or science this 
semester,......<< 

Therefore w = 25

So we have these four facts:

x + y + z + w = 150
x > y
y + z = 80
w = 25

Substituting w = 25 in the first equation,

x + y + z + 25 = 150
     x + y + z = 125

Substitute 80 for y + z in that:

        x + 80 = 125 
             x = 45

Substitute x = 45 in x > y

            45 > y

Solve y + z = 80 for y
          y = 80 - z

Substitute 80 - z for y in 45 > y

          45 > 80 - z

           z > 80 - 45

           z > 35

So the minimum value of z is 36, for
that is the smallest possible whole
number greater than 35.

Edwin